# Rumus Sudut Ganda dari Penjumlahan

$$\begin{align*} \sin(2\alpha) & =\sin(\alpha+\alpha)\\ & =\sin(\alpha)\cos(\alpha)+\cos(\alpha)\sin(\alpha)\\ & =2\sin(\alpha)\cos(\alpha)\\ \cos(2\alpha) & =\cos(\alpha+\alpha)\\ & =\cos(\alpha)\cos(\alpha)-\sin(\alpha)\sin(\alpha)\\ & =\cos^{2}(\alpha)-\sin^{2}(\alpha)\\ & \text{atau}\\ & =\cos^{2}(\alpha)-(1-\cos^{2}(\alpha))\\ & =\cos^{2}(\alpha)-1+\cos^{2}(\alpha)\\ & =2\cos^{2}(\alpha)-1\\ & \text{atau}\\ & =(1-\sin^{2}(\alpha))-\sin^{2}(\alpha)\\ & =1-2\sin^{2}(\alpha)\\ \tan(2\alpha) & =\tan(\alpha+\alpha)\\ & =\frac{\tan(\alpha)+\tan(\alpha)}{1-\tan(\alpha)\tan(\alpha)}\\ & =\frac{2\tan(\alpha)}{1-\tan^{2}(\alpha)} \end{align*}$$

# Kesimpulan

Untuk $\alpha$ dan $\beta$ sebuah sudut apa pun berlaku

$$\begin{align*} \sin(2\alpha)&=2\sin(\alpha)\cos(\alpha)\\ \cos(2\alpha)&=\cos^{2}(\alpha)-\sin^{2}(\alpha)\\ &=2\cos^{2}(\alpha)-1\\ &=1-2\sin^{2}(\alpha)\\ \tan(2\alpha)&=\dfrac{2\tan(\alpha)}{1-\tan^{2}(\alpha)} \end{align*}$$